• August 02, 2021, 08:26:54 AM

### Author Topic:  rational function  (Read 1620 times)

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#### marcm200

• 3e
• Posts: 1013
##### Re: rational function
« Reply #45 on: June 27, 2021, 01:56:33 PM »
My favourite Julia set of lately: Circles on circles on circles... - which reminded me of my drawings in Kindergarten: too thick a pen and a shakey hand

$f(z) := \frac{1}{z^3+2.099609375\cdot z+0.349609375}$

recreated from https://web.archive.org/web/20161024194536/http://www.ijon.de/mathe/julia/some_julia_sets_3.html, a site Adam pointet out.

The image below is a hybrid: Reliably computed is yellow, a covering of the Julia set boundary within the 4-square (plus attracting periodic points and poles), determined by identifying strongly connected components of a one-iteration graph with pixels having area. However, in the rational case, I cannot rule out the existence of more SCCs partially extending outside or lying fully outside.

Numerically, the critical points are computed (small rectangles with guiding turquois vertical lines, starting from image bottom). Rectangles without a point in them are the zeros of the numerator polynomial of the derivative, rectangles with a yellow point within are poles of the denominator polynomial of f (those are algorithmically treated as their own SCC and hence show up here).

Two attracting cycles were numerically identified, both period-2: The horizontal purple line, starting far out right on the x-axis and the vertical red one, joining the two isolated yellow blobs.

The parameters used here differ slightly from the ones on the site, as I prefer working with exactly double-representable numbers by using a near dyadic fraction, hoping the overall structure of the set remains the same (i.e. intersecting Jordan curves).

• Fractal Frankfurter
• Posts: 541
##### Re: rational function
« Reply #46 on: June 27, 2021, 02:17:44 PM »
... exactly double-representable numbers by using a near dyadic fraction

does it  mean a*2^-25 ? ( a is an integer)

stackoverflow question: how-to-tell-if-a-number-is-exactly-representable-as-a-32-bit-ieee-float

How important is to use representable numbers ?  I know that it should increase precision, but double numbers are ( if I'm not wrong) normalised to representable automatically ?

Can you show example that using nonrepresentable numbers changes result ? ( in important way)

« Last Edit: June 27, 2021, 03:31:59 PM by Adam Majewski, Reason: descr »

#### marcm200

• 3e
• Posts: 1013
##### Re: rational function
« Reply #47 on: June 27, 2021, 02:30:20 PM »
does it  mean a*2^-25 ? ( a is an integer)

Exactly. The value on the site said d=2.1, which when read in using e.g. sscanf in C++ will be converted to a close double-representable number, but that conversion relies on the currently set rounding mode (usually TO_NEAREST), which makes the output reproducible only under that rounding mode. I try to avoid those implicit assumptions when working with cell-mapping where I aim at getting some kind of guarantee.

#### marcm200

• 3e
• Posts: 1013
##### Re: rational function
« Reply #48 on: June 28, 2021, 04:11:37 PM »
$z^2+c + \frac{a}{z}$
gives an interesting "threaded" shape (idea from previously mentioned site).

Left is the basilica (a=i*39/2^17), yellow pixels (representing small complex intervals) show the strongly connected components that are fully inside the 2-square (Julia set boundary, attracting periodic points and poles).

Right is a basilica-like example at c=-1+i*1/16, a=1/8 that could consist of small threads (dendrites)? Or maybe it disintegrates into a Cantor set at finer resolutions?

• Fractal Frankfurter
• Posts: 541
##### Re: rational function
« Reply #49 on: June 28, 2021, 05:06:22 PM »

#### youhn

• Fractal Feline
• Posts: 153
##### Re: rational function
« Reply #50 on: June 28, 2021, 05:28:57 PM »
Interesting work!*

Quote
Right is a basilica-like example at c=-1+i*1/16, a=1/8 that could consist of small threads (dendrites)? Or maybe it disintegrates into a Cantor set at finer resolutions?

Since most julia sets are very self-similar, I wouldn't expect 2 different patterns. The image shows a discontinuation in space, when looking at the perceived shapes. Therefore I wouldn't expect thin dendrites. My bet is on cantor-set like.

( * though I don't fully grasp all the math stuff)

• Fractal Frankfurter
• Posts: 541
##### Re: rational function
« Reply #51 on: July 05, 2021, 08:53:06 PM »
Code: [Select]
convert "${b}".pgm -resize 2000x2000 "${b}".png # downscale convert LCM_Fatou_abi.png -morphology Erode Octagon LCM_Fatou_abi_erode_fat-white-on-black.png  # thicken the lines
Analysis of rational map :

Rational function ( map) in 3 forms  :
* f(z)=1/(z3+dz+c) mit c=0,37 und d=2,1 ( original)
* f(z) := 1/{z^3 + 2.099609375*z  +0.349609375} (using representable numbers )
*  f(z) = 512/(512*z^3+1075*z+179)   ( ratsimp , is it useful ?)

First derivative wrt z :

f'(z) = (-3 z^2 - 2.09961)/(z^3 + 2.09961 z + 0.349609)^2

Degree of function : d= 3

Number of critical points = 2*d-2 = 4

I have found 2 critical points :

solve (3*z^2+2.099609375)/(z^3+2.099609375*z+0.349609375)^2=0

[-0.8365822085525526*%i,0.8365822085525526*%i]

but it seems that all 2 fall into one period 2  cycle :

{0.4101296722285251 +0.5079485669960784*I,
0.4101296722285251 -0.5079485669960784*I}

{ +0.1147519899962201 +0.0000000000000000*I ,  +1.6890328811664670 +0.0000000000000000*I }

Also infinity is not attracting point here. Points from the exterior is also attracetd to these 2 cycles

Basins are colored : 2 darker colors  = basin of one cycle, 2 lighter colors =  basin of other cycle
« Last Edit: July 05, 2021, 10:09:41 PM by Adam Majewski, Reason: infinity »

#### marcm200

• 3e
• Posts: 1013
##### Re: rational function
« Reply #52 on: July 17, 2021, 01:28:04 PM »
A weird rational function:
$f_3(z,c) := \frac{z^3+c}{z^2}$
which - for nonzero c - has no finite fix-points - neither attracting nor otherwise.

In maxima:
Code: [Select]
degree:3;f:(z^degree+c)/z^(degree-1);f_fixpoints:solve(f=z,z);
Are there other families of rationals that lack a specific periodic length altogether? (z^d+c)/z^(d-1) never  has a fix point for any choice of positive integer d and complex nonzero c. In a sense it seems that all the fix-points group at c=0 - and then there aren't any left for other c's.

Are there rationals which do not have a parameter set to allow for an attracting cycle of some specific length? In the polynomial Mset z^2+c, any positive integer occurs for an attracting cycle for some c, but maybe that's different for (some) rationals?

• Fractal Frankfurter
• Posts: 541
##### Re: rational function
« Reply #53 on: July 17, 2021, 03:38:50 PM »
...
Are there other families of rationals that lack a specific periodic length altogether? (z^d+c)/z^(d-1) never  has a fix point for any choice of positive integer d and complex nonzero c. In a sense it seems that all the fix-points group at c=0 - and then there aren't any left for other c's.

Are there rationals which do not have a parameter set to allow for an attracting cycle of some specific length? In the polynomial Mset z^2+c, any positive integer occurs for an attracting cycle for some c, but maybe that's different for (some) rationals?
MathOverflow ?

#### xenodreambuie

• Fractal Friar
• Posts: 149
##### Re: rational function
« Reply #54 on: July 18, 2021, 12:25:32 AM »
Interesting. Here is the Julia set for (z3+1)/z2 coloured by log of mean |z|. No points escape or converge. Different values of c just rotate and scale the result. Using inverse iteration instead shows a Newton-like set of similar shape, that leaves big holes at each nexus instead of connecting.

#### marcm200

• 3e
• Posts: 1013
##### Re: rational function
« Reply #55 on: July 18, 2021, 10:59:55 AM »
No points escape or converge.
When I first computed an image, all I got was a black screen - everything bounded, so I went on error-hunting, but couldn't find one. So, nice to see that you got the same result.

Some initial thoughts (not fully worked out):

(z^3+1)/z^2 can be seen as a perturbed linear map z+1/z^2. Iterating e.g. from a point on the positive real axis, z=2, one adds a small positive amount in each step,  so z grows. But how fast? The amount added gets smaller as z grows - but will this lead to divergence or converging to a finite value? (Reminds me of the difference between adding up 1/n and 1/n^2).

0,inf form a 2-cycle. I tried to compute the multiplier of that cycle and got the value 1 (is this correct?), so a parabolic infinity, interesting!

Fix-point-free rational maps can be generated from a polynomial g(z) by f(z) := (z*g(z)+c) / g(z) as by equating f(z)=z all but the c term cancel out after multiplication with the denominator (although I would still need to formally prove that numerator and denominator of f are relatively prime).

In a general setting, one could use f[1]-z=0 <=> g(z)/h(z)-z=0 => g(z)-z*h(z)=0 and by setting all coefficients of this polynomial's z-terms with positive power to zero, solving a system of equations in the initial coefficients of g,h (however one then needs to set a degree bound on g,h at the start).

For 2-cycles, so far I haven't found a solution (or proven non-solubility).

• Fractal Frankfurter
• Posts: 541
##### Re: rational function
« Reply #56 on: July 18, 2021, 11:18:15 AM »
...No points escape or converge. Different values of c just rotate and scale the result. ...
Is it Siegel disc ?

#### xenodreambuie

• Fractal Friar
• Posts: 149
##### Re: rational function
« Reply #57 on: July 19, 2021, 12:37:19 AM »
Is it Siegel disc ?

I wouldn't think so but someone else would need to answer that.

#### xenodreambuie

• Fractal Friar
• Posts: 149
##### Re: rational function
« Reply #58 on: July 19, 2021, 10:42:10 AM »
Here is an image using a torus orbit trap. The torus has outer radius 2 and would sit just inside the first iteration torus (white/teal). The iterations make inner and outer copies, then the third iteration (orange) also has copies around the left and right branching points.

• Fractal Frankfurter
• Posts: 541
##### Re: rational function
« Reply #59 on: July 24, 2021, 08:36:05 PM »
f(z)=1/(z3+dz+c) mit c=0 und d=-3(1+i), dargestellt auf [-3;3]x[-3;3].

2 critical points :   { -0.4550898605622273*I -1.098684113467809,  0.4550898605622273*I+1.098684113467809};
both critical points tend to period 2 cycle  = {0, infinity}

Whole plane ( sphere) is a basin of attraction of period 2 cycle ( which is divided into 2 components )
Julia set is a boundary.

Code in  commons

« Last Edit: July 25, 2021, 11:31:40 AM by Adam Majewski, Reason: descr »

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